What's My Number?

100 mathematicians are in a prison. The guardian chooses to play the following sick game:

• For each mathematician, the guardian draws a random number between 1 and 100, writes it on a hat, and places it on the mathematician.
• All the mathematicians will go to the court, seeing all the numbers except his own.
• All the mathematicians return to their cells, and write down on a paper, what they think their number is.

At the end, if at least one of the mathematicians is right, they all get to go free. If all of them are wrong, they die.

Notice that the mathematicians have no knowledge of their own number, and all numbers are drawn independently, i.e. the numbers could be 1 through 100 or all the numbers could be 17.

The mathematicians can agree on a strategy prior to the beginning of the game. Find a strategy so that they'll get to live.

This is originally taken from Yaniv's Gems.

Solution